Graphics Reference
In-Depth Information
Let
s
f
(
r
). Provethat 0
f
(
n
)
s
f
(
x
)
dx
f
(1).
If lim
f
(
x
)
dx
exists, prove that
f
(
n
) is convergent.
If
f
(
n
) is convergent, prove that lim
f
(
x
)
dx
exists.
(58)
Provethat
1
n
2. Comparewith qn 27.
(59)
Provethat 2
n
2
1
r
2
n
1.
60 (
Abel
, 1828) Provethat
1/(
n
log
n
) is divergent for
n
2.
61 With thenotation of qn 57, lt
D
s
f
(
x
)
dx
.
Provethat
D
D
f
(
x
)
dx
f
(
n
1)
0.
Deduce that (
D
) is convergent. Notice that this result holds
whether
f
(
n
) is convergent or divergent.
Summary
-
series of positive terms
First comparison test
qn 26
If 0
a
b
, for all
n
, and
b
is convergent,
then
a
is convergent. If
a
is divergent then
b
is divergent.
Theorem
qns 30, 32
1/
n
is convergent when
1 and divergent
when
1.
Cauchy
'
s nth root test
qns 35, 38
and (
a
a
k
, then
a
is convergent
when
k
1 and divergent when
k
1.
d'Alembert's ratio test
qns 43, 47
If 0
)
is
convergent when
k
1 and divergent when
k
1.
Second comparison test
qn 53
If 0
a
and (
a
/
a
)
k
, then
a
If 0
a
,0
b
and 0
m
a
/
b
M
, for all
n
, then
a
and
b
are both convergent or
both divergent.
